Identifier
-
Mp00202:
Integer partitions
—first row removal⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001431: Dyck paths ⟶ ℤ
Values
[1,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 0
[2,2] => [2] => [1,0,1,0] => [1,1,0,0] => 1
[2,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 0
[1,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 1
[3,2] => [2] => [1,0,1,0] => [1,1,0,0] => 1
[3,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 0
[2,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
[2,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 1
[1,1,1,1,1] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0] => 1
[4,2] => [2] => [1,0,1,0] => [1,1,0,0] => 1
[4,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 0
[3,3] => [3] => [1,0,1,0,1,0] => [1,1,0,0,1,0] => 1
[3,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
[3,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 1
[2,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 0
[2,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => 2
[2,1,1,1,1] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0] => 1
[1,1,1,1,1,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 1
[5,2] => [2] => [1,0,1,0] => [1,1,0,0] => 1
[5,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 0
[4,3] => [3] => [1,0,1,0,1,0] => [1,1,0,0,1,0] => 1
[4,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
[4,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 1
[3,3,1] => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[3,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 0
[3,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => 2
[3,1,1,1,1] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0] => 1
[2,2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => 1
[2,2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 2
[2,1,1,1,1,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 1
[6,2] => [2] => [1,0,1,0] => [1,1,0,0] => 1
[6,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 0
[5,3] => [3] => [1,0,1,0,1,0] => [1,1,0,0,1,0] => 1
[5,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
[5,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 1
[4,4] => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0] => 1
[4,3,1] => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[4,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 0
[4,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => 2
[4,1,1,1,1] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0] => 1
[3,3,2] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 3
[3,3,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 2
[3,2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => 1
[3,2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 2
[3,1,1,1,1,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 1
[2,2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 0
[2,2,2,1,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 2
[7,2] => [2] => [1,0,1,0] => [1,1,0,0] => 1
[7,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 0
[6,3] => [3] => [1,0,1,0,1,0] => [1,1,0,0,1,0] => 1
[6,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
[6,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 1
[5,4] => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0] => 1
[5,3,1] => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[5,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 0
[5,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => 2
[5,1,1,1,1] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0] => 1
[4,4,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 2
[4,3,2] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 3
[4,3,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 2
[4,2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => 1
[4,2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 2
[4,1,1,1,1,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 1
[3,3,3] => [3,3] => [1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 1
[3,3,2,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 2
[3,2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 0
[3,2,2,1,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 2
[2,2,2,2,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 1
[8,2] => [2] => [1,0,1,0] => [1,1,0,0] => 1
[8,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 0
[7,3] => [3] => [1,0,1,0,1,0] => [1,1,0,0,1,0] => 1
[7,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
[7,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 1
[6,4] => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0] => 1
[6,3,1] => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[6,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 0
[6,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => 2
[6,1,1,1,1] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0] => 1
[5,5] => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 1
[5,4,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 2
[5,3,2] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 3
[5,3,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 2
[5,2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => 1
[5,2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 2
[5,1,1,1,1,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 1
[4,4,2] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 2
[4,3,3] => [3,3] => [1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 1
[4,3,2,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 2
[4,2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 0
[4,2,2,1,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 2
[3,3,3,1] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 1
[3,3,2,2] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 4
[3,2,2,2,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 1
[2,2,2,2,2] => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 1
[9,2] => [2] => [1,0,1,0] => [1,1,0,0] => 1
[9,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 0
[8,3] => [3] => [1,0,1,0,1,0] => [1,1,0,0,1,0] => 1
[8,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
[8,1,1,1] => [1,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 1
[7,4] => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0] => 1
[7,3,1] => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
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Description
Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
Map
Elizalde-Deutsch bijection
Description
The Elizalde-Deutsch bijection on Dyck paths.
.Let $n$ be the length of the Dyck path. Consider the steps $1,n,2,n-1,\dots$ of $D$. When considering the $i$-th step its corresponding matching step has not yet been read, let the $i$-th step of the image of $D$ be an up step, otherwise let it be a down step.
.Let $n$ be the length of the Dyck path. Consider the steps $1,n,2,n-1,\dots$ of $D$. When considering the $i$-th step its corresponding matching step has not yet been read, let the $i$-th step of the image of $D$ be an up step, otherwise let it be a down step.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
first row removal
Description
Removes the first entry of an integer partition
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